Abstract: This paper gives a complete description of ultrametric spaces up to uniform equivalence. It also describes all metric spaces which can be mapped onto ultrametric spaces by a non-expanding one-to-one map. Moreover, it describes particular classes of spaces, for which such a map has a continuous (uniformly continuous) inverse map. This gives a complete solution for the Hausdorff-Bayod Problem (what metric spaces admit a subdominant ultrametric?) as well as for two other problems posed by Bayod and Martínez-Maurica in 1990. Further, we prove that for any metric space , there exists the greatest non-expanding ultrametric image of (an ultrametrization of ), i.e., the category of ultrametric spaces and non-expanding maps is a reflective subcategory in the category of all metric spaces and the same maps. In Section II, for any cardinal , we define a complete ultrametric space of weight such that any metric space of weight is an image of a subset of under a non-expanding, open, and compact map with totally-bounded pre-images of compact subsets. This strengthens Hausdorff-Morita, Morita-de Groot and Nagami theorems. We also construct an ultrametric space , which is a universal pre-image of all metric spaces of weight under non-expanding open maps. We define a functor from the category of ultrametric spaces to a category of Boolean algebras such that algebras and are isomorphic iff the completions of spaces and are uniformly homeomorphic. Some properties of the functor and the ultrametrization functor are discussed.